%0 Journal Article
%T On strongly dense submodulesâ€Ž
%J Bulletin of the Iranian Mathematical Society
%I Iranian Mathematical Society (IMS)
%Z 1017-060X
%A Ghashghaei, E.
%A Namdari, M.
%D 2016
%\ 06/01/2016
%V 42
%N 3
%P 731-747
%! On strongly dense submodulesâ€Ž
%K Strongly essential submodule
%K strongly dense submodule
%K singular submodule
%K special submodule
%K column submodule
%R
%X The submodules with the property of the title ( a submodule $N$ of an $R$-module $M$ is called strongly dense in $M$, denoted by $N\leq_{sd}M$, if for any index set $I$, $\prod _{I}N\leq_{d}\prod _{I}M$) are introduced and fully investigated. It is shown that for each submodule $N$ of $M$ there exists the smallest subset $D'\subseteq M$ such that $N+D'$ is a strongly dense submodule of $M$ and $D'\bigcap N=0$. We also introduce a class of modules in which the two concepts of strong essentiality and strong density coincide. It is also shown that for any module $M$, dense submodules in $M$ are strongly dense if and only if $M\leq_{sd} \tilde{E}(M)$, where $\tilde{E}(M)$ is the rational hull of $M$. It is proved that $R$ has no strongly dense left ideal if and only if no nonzero-element of every cyclic $R$-module $M$ has a strongly dense annihilator in $R$. Finally, some appropriate properties and new concepts related to strong density are defined and studied.
%U http://bims.iranjournals.ir/article_809_9508d8b8a48f740c5b8f741a57040ea9.pdf